Spin angular momentum of the electron: one-loop studies

We combine bare perturbation theory with imaginary time evolution technique to study one-loop radiative corrections to spin angular momentum of the electron. We use for this purpose quantum electrodynamics in covariant and Coulomb gauges. We discuss intricacies of proper implementation of imaginary time evolution as well as the difference between results obtained with Pauli-Villars and three-dimensional cutoff regularizations.


I. INTRODUCTION
The electron, undoubtedly one of the most fundamental constituents of matter, is characterized by a set of physical properties such as the mass, charge, magnetic moment, and spin.
Experimental studies of its mass and charge, m and e below, started in the late nineteenth century in a series of experiments conducted by Thomson [1]. They have been successfully continued ever since. By contrast, progress in theoretical characterization of these parameters is rather uninspiring, if we notice that dimensionless quantities involving them-such as the fine structure constant or ratios of the electron mass to other lepton masses-have never been convincingly estimated.
The electron's intrinsic magnetic moment was introduced by Uhlenbeck and Goudsmit [2] about a century ago in an attempt to explain the anomalous Zeeman effect, which was discovered by Preston at the same time Thomson conducted his electron experiments [3]. Its understanding rapidly progressed soon after thanks to Dirac [4], whose theory predicted e 2m (1) for the electron's magnetic moment. Two decades later [5], Schwinger found a more accurate approximation through a one-loop quantum electrodynamics (QED) calculation replacing (1) with where is the fine structure constant written here in the Heaviside-Lorentz system of units combined with = c = 1 (we use such units throughout this work). This prediction immediately explained spectroscopic "anomalies" found in measurements of Nafe and Nelson [6] and Foley and Kusch [7] that were done concurrently with Schwinger's calculations. Ever since perturbative calculations of the electron's magnetic moment have gone hand in hand with various experimental measurements reaching astonishing accuracy [8]. These efforts allowed for some of the most stringent tests of QED.
The electron's spin was introduced together with its intrinsic magnetic moment in [2]. It was then put on a firm theoretical basis by Dirac [4], whose relativistic quantum mechanics leads to the following expression for the angular momentum operator [9] where ψ is the Dirac field operator, :: denotes normal ordering, and γ are Dirac matrices. The first (second) operator in (4) is the fermionic spin (orbital) angular momentum operator. Consider now the electron at rest, whose spin is polarized in the ±z direction. The expectation value of operator (4), in the corresponding quantum state |Ψ , is s z δ i3 , where reflects the fact that the electron's spin equals one-half. The orbital component of the angular momentum operator does not contribute to such an expectation value and so one finds We will refer to the left-hand side of this equation as spin angular momentum of the electron. The situation is considerably more complex in QED, where the total angular momentum operator, besides fermionic components (4), contains also electromagnetic spin and orbital angular momentum operators. More importantly, expectation values of all these operators receive radiative corrections. It is the purpose of this work to compute such corrections to the right-hand side of (7).
Primary motivation for these studies comes from our interest in fundamental properties of the electron. The expectation value of the fermionic spin angular momentum operator should be in principle measurable due to gauge invariance of such an operator. As a result, it should provide one more insight into the nature of the electron just as, e.g., the electron's magnetic moment does (2).
There are, of course, other reasons for pursuing this line of research. One of them is the angular momentum controversy [10] widely discussed in the context of the so-called nucleon spin crisis [10,11]. While challenges related to nucleon's angular momentum are much bigger on the technical level and quite different from the physical perspective than those encountered in the corresponding electron studies, it seems to us reasonable to make sure that QED calculations are properly done and understood in the first place.
Spin angular momentum of the electron was studied not long ago in [12,13]. These calculations were done in the light-front formalism, employed the light-cone gauge, and used either Pauli-Villars or dimensional regularization. They are, on the technical level, very different from our studies as we use imaginary time evolution formalism, work in covariant and Coulomb gauges, and use either Pauli-Villars or three-dimensional (3D) cutoff regularizations. Therefore, we see our work as complementary to the previous efforts. Among other things, this paper presents some non-trivial results on implementation of imaginary time evolution, it supports gauge independence of the earlier results, and it raises some questions about regularization (in)dependence of QED calculations. Its outline is the following.
We explain in Sec. II the approach that we use to carry out computations. Next, we discuss in Sec. III different contributions to spin angular momentum of the electron. Then, in Sec. IV, we compute one-loop radiative corrections to this quantity in Pauli-Villars-regularized QED. After that, we show in Sec. V that a different result is obtained in the theory regularized with the 3D cutoff. The discussion of these findings is presented in Sec. VI. Several appendices are added to this paper to make its main body better readable and to facilitate verification of our results. We explain our notation in Appendix A and collect all bispinor matrix elements, which we use throughout this work, in Appendix B. The intricacies associated with implementation of imaginary time evolution are discussed in Appendix C, while adaptation of the Pauli-Villars regularization technique to our problem is presented in Appendix D.

II. BASICS
The starting point for our considerations is the QED Lagrangian density where the bare mass and charge of the electron are denoted by m o and e o and the remaining symbols follow all standard conventions. The fermionic spin angular momentum operator, which we have already introduced in Sec. I, can be conveniently written as This operator is gauge invariant. Therefore, we assume that its expectation value is independent of the gauge choice used in the computations (a brief discussion of subtle points associated with gauge independence can be found in Sec. 2.5.2. of [10]). Moreover, as the ground-state expectation value of this operator does not depend on time, we set to simplify the discussion in the intermediate steps (as a self-consistency check, we have verified that z 0 drops out from the final result if it is not set to zero). We will compute expectation value of (9) in the QED ground state with one net electron at rest, which we denote as |Ωs . The term net refers to the fact that besides electrons in electron-positron pairs, there is one electron in such a state. For this purpose, we will use imaginary time evolution technique, which we will briefly summarize below.
It starts with the one-electron ground state of the free Hamiltonian where |0 is the vacuum state of the free theory and the a 0s operator is introduced in Appendix A. State (11) describes the zero-momentum electron in such a spin state s that J i spin• 0s = s z δ i3 . It is then evolved in time (its non-trivial dynamics is induced by the interaction Hamiltonian d 3 x H int ). Enforcement of the imaginary time limit leads to [14] J spin• Ωs = lim where the interaction-picture operators are labeled with the index I, T is the time-ordering operator, and the calculations will be performed with T > 0 until the limit will be taken. The important thing to remember in the following discussion is that total angular momentum in states |0s and |Ωs is the same and equals s z δ i3 , which we will comprehensively discuss in [15].
To compute (12), one needs electromagnetic and fermionic propagators and the interaction Hamiltonian density, all expressed in terms of interaction-picture fields.
The fermionic propagator is gauge invariant and reads On the contrary, expressions for electromagnetic propagators and interaction Hamiltonian densities are gauge non-invariant. We will do calculations in the covariant and Coulomb gauges.
In the general covariant gauge, the gauge-fixing term is added to the Lagrangian density (the term general refers to the arbitrary non-zero value of ξ). Subsequent quantization of the theory leads to which is discussed in [9,16]. Moreover, In the Coulomb gauge, the condition ∂ i A i = 0 is imposed and one ends up with [9] and Since we will be doing the perturbation theory around the one-electron state, we will be encountering the normalizing constant While such a constant is formally infinite, it gets unambiguously cancelled during computations. This happens because all expressions that contribute to the final result describe processes that happen homogeneously in space. As a result, the outermost spatial integral in every such expression is done over a function that is constant in space and so it exactly cancels down normalizing constant (20) appearing in the denominator of such an expression. Needless to say, factors like (20) are frequently encountered in studies involving delocalized states (see e.g. above-cited [10]).

III. PERTURBATIVE EXPANSION
We will derive here an unregularized expression for spin angular momentum of the electron in the general covariant gauge. As will be shown in Secs. IV and V, it is then an easy exercise to see what form it takes in either the Pauli-Villars or 3D cutoff regularization. Moreover, such an expression will be also reused in Coulomb-gauge calculations. It is thus convenient to work initially without any particular regularization scheme.
To proceed, we use the bare perturbation theory expanding (12b) in the series in e o Zeroth-order contribution (21a) is illustrated in Fig. 1. We obtain after using (A4) and (A5) which has been already mentioned below (11). Since we will be discussing diagrams depicted in different figures, we introduce the following notation. The unregularized contribution of the diagram from Fig. X to J i spin• Ωs will be referred to as Diag. X. For example, a trivial illustration of this notation is The rules for drawing position-space Feynman diagrams, which we present in Figs. 1-3, can be deduced without much effort by comparing those diagrams to the analytical expressions that we list for them. There is no need to dwell on these rules because all diagrams will be drawn only after the analytical expressions will be worked out.
To compute (21b), we need the following matrix element that can be obtained through Wick's theorem in combination with (14) and (A4) (22). The grey box stands for Γ i operator (9) and the external lines are for zero-momentum electrons (the same notation is used in all our figures). where comes from contractions on external lines. Matrix element (24) can be additionally simplified with (10) and (25) leading to e if z = 1. Its contractions with the photon propagator are diagrammatically depicted in Fig. 2.
To compute (21c), we proceed similarly as in (24) getting whose contractions with the photon propagator are diagrammatically shown in Fig. 3. Replacements (24g) and (26c) produce a factor of 2 during evaluation of the diagrams, which cancels 1/2! prefactors from (21b) and (21c). To correctly compute contributions of different diagrams to spin angular momentum of the electron, one must properly enforce limit (13). To illustrate the subtle point here, we note that integration over time leads to expressions of the form Limit (13) cannot be taken on (27). The standard textbook solution of this complication is to transfer the −i0 from the limit to the imaginary part of propagators' denominators. After that, the limit T → ∞ is taken. This leads to the Dirac delta function due to the following well-known identity Such a procedure, greatly simplifying calculations, leads to incorrect results when Diags. 2b, 2c, and 3a are considered. To overcome this difficulty, we will simplify expressions containing (27) up to the point, where limit (13) can be taken. Key technical results regarding this procedure are explained in Appendix C. They are used in this section under tacit assumption that some infrared regularization will be implemented later on to facilitate enforcement of imaginary time limit (13). Moreover, to make equations a bit more compact, we introduce the following notation Diagram 3a. We start computation of diagrams with where and we have employed identities (B1) and (B2). Note that we only list those arguments of the function F that are most relevant for enforcement of the imaginary time limit. Using (C8), we get We mention in passing that term (33b) cannot be obtained without careful treatment of factors like (27); see the discussion in Appendix C. The procedure described between (27) and (28) produces only ill-defined term (33a). Diagrams 2b and 2c. Now, we compute Before moving on, we note that the procedure outlined between (27) and (28) leads to δ(f − q) producing the meaningless factor of 1/i0 in the above expression due to (25). Employing (B3)-(B4), (34) can be written as (35) Then using (C15), we find Computation of With the help of (B5), (B6), and (C21) we arrive at We mention in passing that the procedure discussed between (27) and (28) gives a correct result here.
There are no other one-loop contributions to spin angular momentum of the electron in covariantly quantized QED. Indeed, the disconnected vacuum bubble Diags. 2d and 3b immediately cancel out due to the difference in overall signs of (21b) and (21c). Therefore, there is no need to write down expressions for them. Moreover, both vanish. We will skip the discussion of these two not-so-interesting computations as they can be straightforwardly carried out along the lines of other calculations from this work. The final unregularized result for spin angular momentum of the electron comes from Diags. 1, 2a-2c and 3a which can be obtained by adding (22) and (40) to Note that the offending, linearly divergent in T , terms cancel out in (44).
It is now important to stress that the resulting expression for spin angular momentum of the electron does not have a definite value until some regularization scheme is specified. We will illustrate this fact by showing that distinct finite values for (43) can be obtained in Pauli-Villars (Secs. IV) and 3D cutoff (Sec. V) regularizations.

IV. PAULI-VILLARS REGULARIZATION
We will determine in this section spin angular momentum of the electron in the Pauli-Villarsregularized theory. Application of this regularization to our problem is discussed in Appendix D. The small photon mass λ is introduced there to regulate the infrared (IR) sector of the calculation while the large mass Λ of Pauli-Villars ghost particles is used to take care of the ultraviolet (UV) part of the problem The calculations will be done in the Feynman gauge which is something one must keep in mind when comparing the following expressions to the ones from Sec. III. Such a covariant gauge choice simplifies the discussion a bit (arbitrary-ξ studies will be presented in the next section). Next, we introduce as the Pauli-Villars-regularized version of unregularized Diag. X from Sec. III. Note that limit (13) is already taken in (47). It now follows from the discussion in Sec. III and Appendix D that Moreover, we obtain Diag. 2b| λΛ + Diag. 2c| λΛ + Diag. 3a| λΛ Note that one does not end up here with under the integral. Besides trivial (22), these are the only contributions to the final result. They can be compared to (40) and (44). We can now cast these expressions into a familiar form using the standard procedure. To this aim, we introduce and make use of the Wick rotation technique to get and We start discussing these expressions by noting that it can be now easily verified through [14] that (54) is equal to s z δ i3 (Z 2 − 1), where Z 2 is defined in any textbook on QED.
Next, we take the limits of λ → 0 and Λ → ∞ on the sum of (22), (53), and (54), which leads to Then, we integrate the first term of the integrand in (55b) by parts, which leads to the following compact expression Such a step is allowed because the two functions under the integral sign in (55b) are separately integrable. Integral (56) can be also safely split into two finite integrals. It is then a straightforward exercise to show that thereby lim Λ→∞ λ→0 regardless of the order in which the limits are taken. This means that spin angular momentum of the electron, obtained via the Pauli-Villarsregularized calculation, is where the right-most value comes from replacing the bare electron charge by the physical one. Two remarks are in order now. First, we would like to stress that (59) can be obtained in the bare perturbation theory only after careful implementation of imaginary time evolution (Appendix C).
Second, the same result was derived not long ago in a very different way in [12,13].

V. 3D CUTOFF REGULARIZATION
We will compute in this section spin angular momentum of the electron in the 3D cutoff regularization. The calculations will be done in the general covariant gauge in Sec. V A and in the Coulomb gauge in Sec. V B.
The IR cutoff λ c and the UV cutoff Λ c will be imposed on 3-momenta in either electromagnetic or fermionic propagators. We have the following options for replacements in expressions for propagators where θ is the Heaviside step function.
Replacements (61) and (63) will not be applied to the fermionic propagator. It is so because they exclude the zero-momentum fermionic mode, which is crucially important in our calculations (imaginary time evolution starts with the electron at rest). As can be easily checked, the IR cutoff imposed on the fermionic propagator would lead to vanishing contributions from Diags. 2b and 2c, which would have disastrous consequences for the whole calculation. Therefore, the IR cutoff will be imposed on the electromagnetic propagator. We adopt below the notation from Sec. IV, so that the cutoff-regularized version of unregularized Diag. X from Sec. III will be denoted as The regularization will be eventually removed by taking the limits A. Covariant gauge The following calculations are regularized by imposing IR and UV cutoffs on the electromagnetic propagator and keeping the fermionic propagator unregularized (the additional UV cutoff applied to the fermionic propagator would lead to the same final result, but it is unnecessary). Alternatively, one may impose the IR (UV) cutoff on the electromagnetic (fermionic) propagator. Both choices lead to the same results.
As can be quickly checked by going through the calculations in Sec. III, the cutoff-regularized versions of (40) and (44) are obtained by replacing d 4 p with [d 4 p] in these expressions. Integrating them over p 0 , we end up with and Diag. 2b| λcΛc + Diag. 2c| λcΛc + Diag. 3a| λcΛc Using these results, we get upon taking (65) where again (60) has been used. Quite importantly, this result is ξ-independent. Several remarks are in order now. First, after adding (66) and (67) the resulting integral is IR finite and so λ c can be set to zero then. (68) is obtained after doing the angular and radial integrations first and then taking the limit Λ c → ∞. Alternatively, one may start with angular integrations and then carry out the radial integration with Λ c = ∞. The resulting expression, after rescaling the frequency ω p by m o , reads It can be easily computed after the following change of variables where the integral is trivially equal to one. As a curiosity, we mention that substitution (70) can be found in the Ramanujan's first quarterly report prepared for the University of Madras [17]. Naturally, it has been invoked in other contexts as well (see e.g. Sec. 2.25 of [18]). We find it quite useful in our studies of angular momentum of the electron. Second, unless ξ is fine-tuned, (66) and (67) are (without regularization) logarithmically divergent in both UV and IR. For ξ = ∞, the Landau gauge, these expressions are still IR divergent but UV finite. Moreover, after removing the terms proportional to lim T →∞(1−i0) T , all individual diagrams are UV finite in such a gauge. For ξ = 1/3, the Fried-Yennie gauge, (66) and (67) are IR finite but UV divergent, which reflects the well-known fact that QED in such a gauge has significantly improved IR properties with respect to other covariant gauge choices. In fact, for ξ = 1/3 all individual diagrams are IR finite (the same can be said about integrands in these diagrams).
Last but not least, (68) does not agree with (59). It is thus reasonable to double check it by doing a calculation in a different gauge.

B. Coulomb gauge
We will impose here the IR cutoff on the electromagnetic propagator and the UV one on the fermionic propagator. The following calculations will be done in two independent steps because the two terms in Hamiltonian density (19) will never come together in the one-loop perturbative expansion. So, we write where S ⊥ and S will be called the transverse and longitudinal contributions to J spin• Ωs . To obtain the regularized expression for S ⊥ , we need to replace H I int (x) in (21b) and (21c) by e o : ψ I (x)γ m ψ I (x) : A I m (x), compute resulting expressions with regularized propagators, and take limit (13). All these steps can be easily accomplished with results from Sec. III.
On the other hand, a new calculation has to be done for S , which after regularization reads where is computed with the UV-modified expression for the fermionic propagator. Note that such an expression does not depend on the electromagnetic propagator and so it does not rely on the IR cutoff λ c . Lack of IR regularization does not cause problems here because the longitudinal contribution is IR finite and imaginary time evolution does not need the IR cutoff for its evaluation (Appendix C). Transverse contribution. The Coulomb-gauge results are obtained by replacing covariantgauge electromagnetic propagator (16) with Coulomb-gauge propagator (18) in all equations in Sec. III. For example, the Coulomb-gauge version of Diag. 3a is If we now use cutoff-modified propagators to compute such an expression, we will get Diag. 3a| Coulomb λcΛc (the similar notation will be used below). With the help of matrix elements (B7) and (B8), it is then easy to show that Diag. 2b, 2c, 3a| Coulomb wherek is given by (30) Moreover, using (B9), we find that As other diagrams do not contribute to S ⊥ , it is given in the regularized form by After simple calculations, we get which is UV divergent but IR finite when (65) is enforced. In fact, all individual diagrams in (79) are IR finite after removing regularization (integrands in expressions defining them also have such a property). This is rather unsurprising as many other examples of IR finite QED calculations in the Coulomb gauge can be found in literature (see e.g. [19,20]). Longitudinal contribution. We evaluate now S i | Λc starting with where y = (x 0 , y) and 0s|T : ψ I (z)Γ i ψ I (z) :: with cutoff (62) imposed on the fermionic propagator. (82) is trivially obtained from (24). Spin-independent term (82b) does not contribute to (81). This can be shown, by proceeding with the calculations similarly as below, with whereq is defined in (30). Thus, we are left with .
To integrate out the Coulomb potential, we need the standard trick to justify commutation of spatial and momentum integrals. Namely, where the placeholder stands for the p-dependent function in our calculations. Two remarks are in order now.
First, the above ǫ-regularization can be used to justify commutation of spatial and momentum integrals that we routinely do in all our calculations integrating out the exponential terms to get the Dirac delta functions. This simplifies expressions and takes care of 3-momentum conservation in every vertex. Second, the final result of these calculations is the same if we set ǫ = 0 in the last expression in (85), which we do below to simplify equations. Such a conclusion follows from the fact that S i is IR finite with ǫ = 0.
Moving on, we arrive after simple manipulations at We get with the help of (B10) and (C25) It is now easy to see that the sum of (80) and (87) is equal to the sum of (66) and (67) after taking limit (65). This means that we have obtained in the 3D cutoff regularization the same result for spin angular momentum of the electron in Coulomb and general covariant gauges.

VI. DISCUSSION
We have teamed bare perturbative expansion with the imaginary time evolution technique to study radiative corrections to spin angular momentum of the electron. This required careful implementation of the latter procedure, which we have comprehensively discussed in Appendix C containing results that can be useful in other studies as well.
Using Pauli-Villars regularization, we have rederived in a very different way recent results for spin angular momentum of the electron [12,13], thereby showing equivalence of the light-cone and covariant (Feynman) gauge calculations of this observable. As is discussed in Sec. 2.5.2 of [10], the issue of gauge independence is quite non-trivial and so such explicit verification of the earlier results shall be of interest.
We have then found that a different result is obtained when the 3D cutoff regularization is used. This raises the questions of which result is correct and what is the fundamental reason for disagreement.
Having two different results, (59) and (68), one may try to eliminate one of them with some selfconsistency check. The obvious choice here is to consider total angular momentum of the electron, whose expectation value should be exactly equal to Such a check can verify the overall consistency of the approach used for getting spin angular momentum of the electron. We computed total angular momentum of the electron in 3D cutoffregularized QED that was covariantly quantized [15]. We found that one-loop radiative corrections to its different components cancel out so that (88) is obtained. The same value of total angular momentum of the electron was reported in [12]. These computations, however, were criticized in [13], and so their recalculation may be appropriate. We are thus left with the obvious option that the choice of regularization is responsible for the difference between (59) and (68). The former result is obtained in either Pauli-Villars or dimensional regularization ( [12,13] and Sec. IV), while the later one in the 3D cutoff regularization (Sec. V). Therefore, it is reasonable to compare these regularizations in the context of our studies.
We start by noting that each result was obtained in different gauges. Indeed, (59) was obtained in light-cone and covariant (Feynman) gauges, while (68) was obtained in Coulomb and general covariant gauges. As a result, none of the above-mentioned regularizations seems to break gauge independence of QED.
The main advantage of the 3D cutoff regularization is that it is physically motivated unlike dimensional and Pauli-Villars regularizations. Such motivation comes from the observation that QED cannot describe particles having arbitrarily high energy. Therefore, it seems reasonable to stay within the regime of its validity by bounding their energies with the 3D UV cutoff (the 3D IR cutoff could be also physically justified if one assumes that the Universe is finite). This is particularly well seen in the framework of the old-fashioned perturbation theory, where contributions of excited states, which are always on-shell, are "weighted" by the magnitude of 3-momentum.
The main technical incentive to use the 3D cutoff regularization instead of the dimensional regularization comes from the fact that employment of the latter is problematic in our studies. It is so because one has to deal with objects such as ε µνρσ resisting straightforward extensions to arbitrarily dimensional space-times (Appendix B and e.g. Appendix B.2 of [21]). As a result, calculations are more appealing in the four-dimensional space-time, where all fields and operations on Dirac matrices are unambiguously defined.
The 3D cutoff regularization, just as any other regularization scheme, has its own problems. They can be labeled as either technical or physical and the question is whether they influence computations of spin angular momentum of the electron.
The technical problems follow from the fact that it breaks translational invariance in momentum space, which can complicate evaluation of integrals. Moreover, it can lead to boundary terms during partial integrations. Neither the dimensional nor Pauli-Villars regularization seems to face such problems. In our calculations in Sec. V, however, we do not experience them.
The physical problem with the 3D cutoff regularization is that it breaks Lorentz symmetry unlike Pauli-Villars and dimensional regularizations. The lack of Lorentz symmetry during calculations, however, does not necessarily mean that the final result, which is obtained after removing the regularization, is incorrect. Nonetheless, we suspect that this could be the fundamental reason for the disagreement between (59) and (68). Therefore, it is our educated guess that the former of these two results is correct.
On account of all these remarks, we hope that our work will trigger some discussion about regularization (in)dependence of QED calculations in general and the 3D cutoff regularization in particular. We also hope that our work will raise interest in the experimental studies of spin angular momentum of the electron. Given the fact that components of angular momentum of nucleons are experimentally studied [10,11], we are hopeful that such a quantity can be also measured.
We define contractions of ψ I on zero-momentum external lines as ψ I (x)|0s = u s (2π) 3/2 e −if x , 0s|ψ I (x) = u s (2π) 3/2 e if x , u s = u(0, s), where |0s and f are given by (11) and (25), respectively. The u s bispinors are eigenstates of the z-component of the one-particle fermionic spin angular momentum operator Finally, we mention that there is no summation over s in matrix elements u s · · · u s .

Appendix B: Bispinor matrix elements
The results presented below are obtained in the standard (Dirac) representation of γ matrices. It is then a simple exercise to show that the same results are obtained in all representations unitarily similar to the standard one (Weil, Majorana, etc.). This statement is equivalent to saying that they are invariant under γ µ → U γ µ U † and u s → U u s transformations, where U is an arbitrary unitary matrix of dimension four (see [23,24] for the discussion of representation-independence of various results associated with the Dirac equation).
The following expressions are used in our computations: whereq is given by (30). It is interesting to note that s z -dependence in all the above u s · · · u s matrix elements comes from expressions that critically depend on the 4-dimensional Levi-Civita symbol, whose extension to a d = 4 dimensional space-time, used in the dimensional regularization, is problematic (see e.g. Appendix B.2 of [21]). This can be proved by combining the following easy-to-verify identities u s γ µ u s = η µ0 , (B11) u s γ µ γ ν u s = η µν − 2is z ε 0µν3 , (B12) u s γ µ γ σ γ ν u s = η µσ η ν0 + η σν η µ0 − η µν η σ0 − 2is z ε µσν3 , u s γ 0 γ 1 γ 2 γ 3 u s = 0 (B14) with the observation that any product of γ matrices can be always reduced to the single term containing at most four γ matrices, whose indices are distinct.
which, with the help of (C1), can be cast into the following form χ = π dk 0 G(k 0 , k 0 ) Adopting the procedure described around (C7), and noting that this time G(k 0 , p 0 ) has poles parameterized by M = M ′ = m o (C3), we readily realize that the last two integrals vanish as T → ∞(1 − i0). So, we find that χ = π dk 0 G(k 0 , k 0 ). (C25) Using (28), one immediately finds that the same result is obtained by taking T → ∞ under integral (C22).